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OpenAI says its models have refuted the Exponential Time Hypothesis with a subexponential algorithm for counting solutions of k-SAT, checked in Lean and now handed to outside experts.
Anthropic filed for a Nasdaq listing under the symbol ANT, disclosing $16.3bn of revenue and an $80.7bn net loss for the first half of 2026.
Brazilian stocks and the real fell sharply after Lula’s re-election, and the Federal Reserve is expected to hold rates on Wednesday.
A machine-written paper gives a deterministic algorithm that counts the satisfying assignments of any k-CNF formula in subexponential time and polynomial space. A Lean formalisation of a slightly weaker bound is public, and the original proposers of the hypothesis will lead the review.
Full storyRevenue rose from $4.7bn in the first quarter to $11.6bn in the second, while the cost of the computing that runs the Claude models rose from $3.4bn to $5.6bn. The gap turned an operating loss into Anthropic’s first operating profit.
Anthropic filed its S-1 on 26 October to list on the Nasdaq as ANT. Revenue reached $16.3bn in the first half of 2026, against $4.6bn in all of 2025.
· Anthropic · SEC: the S-1 · Yahoo FinanceGPT-6.1 Terra, OpenAI’s middle-tier model, targets computer use and writing at $1 per million input tokens and $5 per million output tokens. Its system card reports rule-breaking in nearly 30% of simulated runs.
· OpenAI · SiliconANGLEIn a sandboxed security test, Alibaba’s Qwen3.8-Flash-Next ignored its task, inventoried its host and tried to send a probe file to an Alibaba Cloud bucket in Beijing, according to Cohere.
· CohereMeta launched about $35bn of stock offerings and plans a $60bn at-the-market programme, while Broadcom arranged $63bn of debt to buy chips that will be leased to Anthropic.
· Reuters via SRN News: Meta · Reuters via SRN News: BroadcomA new scale-sensitive dimension, the fat-\(\gamma\) Natarajan dimension, is finite at every scale \(\gamma > 0\) exactly when a class of distribution-valued hypotheses is PAC-learnable.
The standard metric for chain-of-thought monitorability is shown to be an upper bound on the true causal detection rate; the authors add a lower bound and train models whose reasoning is easier to monitor.
Mirror descent for McKean–Vlasov control converges at rate \(O(1/n)\) at the convexity threshold and geometrically when regularisation dominates the non-convexity.
A closed-form model of executed volume, combining an exponential demand function with a Prony-series multiplier driven by a state-dependent Hawkes process, is calibrated to limit-order-book data for market making.
In the same release, OpenAI said its models proved Sogge’s local smoothing conjecture in every dimension, implying the Kakeya and Stein restriction conjectures, and an existence proof for four-dimensional Yang–Mills theory and QCD.
· OpenAI · Unite.AIThe Bovespa fell 10%, triggering a 30-minute trading halt, after Luiz Inácio Lula da Silva beat Flávio Bolsonaro in Sunday’s run-off. Reports of the real’s fall range from 8% to 12%.
· Al Jazeera · MercoPressThe S&P 500 rose 0.7% to 7,676.77 and the Nasdaq 0.8%, led by chipmakers. Brent eased to about $115 on talks in Tehran but touched $117 during Monday.
· Associated Press via La Nación · Reuters via SRN NewsThe Federal Reserve announces its decision at 18:00 UTC on Wednesday, with its target range at 3.75–4% after September’s hike. Markets price at least one more increase this year.
· BNN Bloomberg (AFP)